Properties of Toeplitz operators onanalytic function spaces: from function

symbols to distributions

Antti Perala

Department of Mathematics and StatisticsUniversity of Helsinki

Academic dissertation

To be presented for public examinationwith the permission of the Faculty of Science of the University of Helsinkiin Auditorium PIII of Porthania on 7 November 2011 at 12 o’clock noon.

ISBN 978-952-10-7264-2 (Paperback)ISBN 978-952-10-7265-9 (PDF)http://ethesis.helsinki.fi

Unigrafia OyHelsinki 2011

iii

Contents

Acknowledgements vList of contributed articles vii1. Introduction 11.1. Preliminaries 11.2. Boundedness and compactness of Toeplitz operators 31.3. Fredholm theory 72. Properties of Toeplitz operators on analytic function spaces: from

function symbols to distributions 112.1. Article A: Toeplitz operators with distributional symbols on

Bergman spaces 112.2. Article B: New results and open problems on Toeplitz operators

in Bergman spaces 132.3. Article C: Toeplitz operators with distributional symbols on

Fock spaces 152.4. Article D: Toeplitz operators on Bloch-type spaces and classes

of weighted Sobolev distributions 162.5. Article E: A note on the Fredholm properties of Toeplitz

operators on weighted Bergman spaces with matrix-valued symbols 18

3. Notes 213.1. Complementary material 213.2. Errata 21References 23

v

Acknowledgements

The work towards this doctoral thesis was financially supported by severalsources. I am indebted to the Vaisala foundation of the Finnish Academy ofScience and Letters and to the Finnish National Graduate School in Math-ematics and its Applications. My work was also partially supported by theAcademy of Finland project ”Functional analysis and applications”. I wishto express my most sincere gratitude to the above instances for their supportas well as all the people involved.

During the writing process of this dissertation I have had the pleasureof interacting with some of the most influential mathematicians of the field.In particular, I am grateful to Professor Kehe Zhu from SUNY Albany forseveral enlightening discussions and support.

I want to express my gratitude to the pre-examiners of this work, Pro-fessor Mikael Lindstrom and Docent Jouni Rattya. Their suggestions andnumerous pieces of advice have significantly improved the readability of thisthesis.

I am thankful for my advisors Docent Jari Taskinen and Docent JaniVirtanen. Throughout this journey, their guidance has been patient and in-spiring, and I feel infinitely fortunate for the opportunity to be a student ofsuch excellent teachers and brilliant mathematicians. Finally, I wish to thankthe family of J. Virtanen for all the hospitality during my several visits overthe years.

Helsinki, August 2011

Antti Perala

vii

List of contributed articles

This doctoral dissertation consists of an introductory part, the following fiverefereed journal articles and a notes section including some complementaryresults and errata. The author acknowledges the respective journals as theoriginal publishers of these articles.

[A] A. Perala, J. Taskinen, J. Virtanen, Toeplitz operators with distribu-tional symbols on Bergman spaces. Proceedings of the Edinburgh Math-ematical Society, Volume 54, Issue 02, 505–514, (2011).

[B] A. Perala, J. Taskinen, J. Virtanen, New results and open problems onToeplitz operators in Bergman spaces. New York Journal of Mathemat-ics, Volume 17a, 147–164, (2011).

[C] A. Perala, J. Taskinen, J. Virtanen, Toeplitz operators with distribu-tional symbols on Fock spaces. Functiones et Approximatio, Commen-tarii Mathematici, Volume 44, Number 2, 203–213, (2011).

[D] A. Perala, Toeplitz operators on Bloch-type spaces and classes of weightedSobolev distributions. Integral Equations and Operator Theory, Volume71, Number 1, 113–128 (2011).

[E] A. Perala, J. Virtanen, A note on the Fredholm properties of Toeplitzoperators on weighted Bergman spaces with matrix-valued symbols. Op-erators and Matrices, Volume 5, Issue 1, 97–106, (2011).

1

1. Introduction

Toeplitz operators are among the most important types of concrete operatorswith applications to several branches of mathematics including mathemati-cal physics, complex analysis, theory of normed algebras, dynamical systems,random matrix theory and operator theory. They have been even used in areassuch as control theory and fluid dynamics. Despite their apparent simplicity,Toeplitz operators often provide non-trivial examples and counter-examplesto many phenomena of mathematics. A good account on applications is pro-vided in the books [7], [8] and [28].

In this thesis we study boundedness and compactness criteria for Toeplitzoperators by extending the definition from function symbols to distributionalones. This kind of approach takes into account the behaviour of the deriva-tives of analytic functions in a convenient way. We will also look at theFredholm properties of Toeplitz operators with matrix-valued symbols andstudy their index theory.

We will start by reviewing what is generally known about the bounded-ness, compactness and Fredholm theory of Toeplitz operators having functionsymbols. We will discuss Bergman, Fock and Bloch-type spaces, but the mainfocus is on the Bergman space case. The reasons for this are the following: theBergman space setting is the archetypal setting for our purposes, while theBloch and Fock space results are often analogous when appropriate (Gauss-ian, polynomial or logarithmic) weights are applied. On the other hand, mostof our results are for Bergman spaces, anyway. An interested reader can findplenty of material on Bloch and Fock spaces in the references. Our approachto the distributional symbol case is explained in the second part of this thesis.

We have taken the liberty to combine several results from one sourceto one theorem. This does not always reflect the course of thinking leadingto the results, but makes the exposition more readable. Detailed referencesto the corresponding partial results are provided for the convenience of thereader.

The notation is standard: by D be mean the open unit disk of thecomplex plane C, centered at the origin, and T is used for the boundary ofD, T = ∂D. More notation is explained along the way.

1.1. Preliminaries

We will work either on Bergman, Bloch or Fock spaces. For general referenceon these spaces, we mention [14, 36, 37] for Bergman spaces, [14, 35, 36, 37]for Bloch spaces and [12, 15, 18] for Fock spaces. Let 0 < p < ∞, then theBergman space Ap of the unit disk consists of those analytic f : D → Csatisfying

‖f‖p :=

(∫D|f(z)|pdA(z)

)1/p

<∞,

the space H∞ consists of bounded analytic functions on D. When p ≥ 1,we can equip these spaces with respective Lp-norms making them Banachspaces.

2

The Bergman projection P is an integral operator induced by the Bergmankernel Kz;

Pf(z) =

∫Df(w)Kz(w)dA(w) =

∫Df(w)(1− zw)−2dA(w) , z ∈ D. (1.1)

The projection P is bounded Lp → Ap whenever p ∈ (1,∞). It is not boundedL1 → A1 or L∞ → H∞, but the definition still makes sense. If a ∈ L∞, wecan define the Toeplitz operator with symbol a by

Taf(z) = P (af)(z) =

∫D

f(w)a(w)dA(w)

(1− zw)2, z ∈ D. (1.2)

Here f can be assumed to be a member of Ap for 1 ≤ p <∞ or H∞.

If 1 < p < ∞, then the dual space of Ap can be identified with theBergman space Aq where q is the Holder conjugate of p; that is, 1/p+1/q = 1.The dual space of A1 can be identified with the Bloch space B consisting ofthose analytic f : D→ C for which the Bloch seminorm:

‖f‖∗ := supz∈D

(1− |z|2)|f ′(z)|

is finite. Equipped with the norm

‖f‖B = ‖f‖∗ + |f(0)| , f ∈ B,

the Bloch space becomes a Banach space. The Bergman projection (1.1) isbounded from L∞ onto B. The formula (1.2) makes sense also when f ∈ B.We will also consider the general Bloch-type spaces Bd for d > 0, they aredefined similarly to B by using the semi-norm

‖f‖∗,d := supz∈D

(1− |z|2)d|f ′(z)|.

It is possible to define Toeplitz operator on Bd, but then one has to be careful;for large values of d the condition a ∈ L∞ is not sufficient for Ta to be defined.

Let 1 ≤ p < ∞ and γ > 0. Then the Fock space F pγ of the complexplane consists of all entire functions f for which

‖f‖p,γ :=

(∫C|f(z)|pe

−γp|z|22 dAp,γ(z)

)1/p

is finite. Here dAp,γ(z) = cp,γdA(z) is the Lebesgue area measure normalizedso that ∫

Ce−(γp/2)|z|2dAp,γ(z) = 1.

The space F∞γ consists of all entire functions for which the norm

‖f‖∞,γ := supz∈D|f(z)|e

−γ|z|22

3

is finite. The Fock spaces closed subspaces of the respective Lpγ spaces definedin the usual way. The Fock projection P is then the integral operator inducedby the Fock kernel Kz as follows

Pf(z) =

∫Cf(w)Kz(w)e−γ|w|

2

dAp,γ(w)

=

∫Cf(w)eγzwe−γ|w|

2

dAp,γ(w) , z ∈ C.

We note that since we deal with the Bergman and Fock spaces separately,there is no confusion in the notation. The Fock projection is known to bebounded Lpγ → F pγ for each p ∈ [1,∞]. If a ∈ L∞, we can then define theToeplitz operator analogously to (1.2) by the formula

Taf(z) = P (af)(z) =

∫Ca(w)f(w)eγzwe−γ|w|

2

dAp,γ(w) , z ∈ C. (1.3)

1.2. Boundedness and compactness of Toeplitz operators

Boundedness and compactness are among the most fundamental propertiesof linear operators. We recall that an operator T between Banach spaces Xand Y is bounded if T (E) ⊂ Y is bounded for each bounded E ⊂ X. Thenorm of T is then ‖T‖ = supx∈X,‖x‖=1 ‖Tx‖. The operator T is compact if

the closure of T (E) is compact in Y .

Hardy spaces. Suppose that 1 < p <∞, Hp(T) ⊂ Lp(T) is the Hardy spaceand P : Lp(T)→ Hp(T) is the Riesz projection. If a is a measurable function,Ma is the related multiplication operator and Ta is the Toeplitz operator:

Taf = PMaf = P (af),

then Ta is bounded if and only if a ∈ L∞(T). Moreover, Ta is compact if andonly is a is the zero function. Theory of Toeplitz operators on Hardy spacesis documented in several sources, we mention [8] and [13].

Although the boundedness and compactness questions have been settledin the Hardy space case, they are still open in many other spaces, includingthe Bergman, Bloch and Fock spaces. We mention that finite rank Toeplitzoperators on the Bergman space setting were recently characterized by D.Luecking in [20].

Carleson measures. Carleson measures have been invaluable tool in severalaspects of mathematical analysis. They are named after L. Carleson whodeveloped them in Hardy space setting in the course of his famous proof ofthe Corona theorem, see [9]. The boundedness and compactness of Toeplitzoperators can be approached via the use of similar measures in other functionspaces.

A non-negative Borel measure µ on D is said to be Carleson measurefor Bergman spaces if there exists C > 0 such that∫

D|f(z)|pdµ(z) ≤ C

∫D|f(z)|pdA(z)

4

holds for every f ∈ Ap. Equivalently, the embedding ip : Ap → Lp(µ) isrequired to be bounded. Here p ∈ [1,∞) can be chosen freely and we alwaysget the same collection of measures. Denote by β the Bergman metric of Dand write Bβ(z, r) := {w ∈ D : β(z, w) < r} for the Bergman disk of radiusr > 0 centered at z ∈ D. Geometric characterizations for Carleson measureshave been studied by several authors, including W. Hastings, D. Luecking,V. L. Oleinik, B. S. Pavlov and K. Zhu, see [16, 19, 21, 33]. In what follows,we refer to the work of Zhu.

Proposition 1.2.1. (Theorem 7 of [33]) Let r > 0. A non-negative Borel mea-sure µ on D is a Carleson measure on Bergman spaces if and only if thereexists C > 0 such that

µ(B(z, r)) ≤ C|Bβ(z, r)|

for every z ∈ D.

Here, and in what follows, we use |A| to denote the normalized Lebesguemeasure of A. Note that by the virtue of above proposition, the radius r >0 can be chosen freely and we always end up with the same collection ofmeasures.

We can also define vanishing Carleson measures on the Bergman spacesby requiring that the embedding ip is compact for some (and thus all) p ∈(1,∞). Note that p = 1 is not included in the definition. A characterizationusing p = 1 is also possible, see [37]. Again, being vanishing Carleson measureon Bergman spaces is characterized by the following:

Proposition 1.2.2. (Theorem 11 of [33]) Let r > 0. A non-negative Borelmeasure µ on D is a vanishing Carleson measure on Bergman spaces if andonly if

lim|z|→1−

µ(Bβ(z, r))/|Bβ(z, r)| = 0.

Again, every choice of r > 0 in the above proposition will result in thesame collection of measures.

In the above proposition the sets Bβ(z, r) can be replaced by squaresto be defined in (1.4); these squares are actually often referred as Carlesonsquares. It is often up to the person using Carleson measures whether to useBergman disks or Carleson squares; the former provides conformal invariance,the latter is easier when integrating in polar coordinates.

Carleson measures can be defined in Fock spaces, as well. The definitionis analogous, the Bergman disks are replaced by Euclidean disks and theCarleson squares by ordinary ones.

It is known that a non-negative measure will generate bounded Toeplitzoperator on Ap (1 < p < ∞) if and only if it is a Carleson measure onBergman spaces. A result of similar nature is true about Fock spaces. Also,compactness is equivalent to being vanishing Carleson on the respective space.We refer the reader to the sources [18, 19, 23, 33, 37] for detailed analysisregarding Carleson measures. If a is a non-negative integrable function, it is

5

natural to ask whether the measure a(z)dA(z) is Carleson. The same goes forthe Fock spaces and the non-negative Borel measures on the complex plane.

For locally integrable functions of the unit disk, a more general resultwas recently obtained in [26]. Denote by D the family of sets D := D(r, θ)defined as

D = {ρeiφ : r ≤ ρ < 1− (1− r)/2, θ ≤ φ ≤ θ + π(1− r)}. (1.4)

Given a ∈ L1loc, D ∈ D and ζ = ρeiφ ∈ D we denote

aD(ζ) :=1

|D|

∫ ρ

r

∫ φ

θ

a(seiψ)sdψds. (1.5)

For D ∈ D we also denote by d(D) the distance from the set D to theboundary of the unit disk. The main theorems of [26] are as follows:

Theorem 1.2.3. (Theorem 2.3 and Theorem 2.6 of [26]) Let 1 < p < ∞ anda ∈ L1

loc. Suppose that there exists C > 0 such that

|aD(ζ)| ≤ C (1.6)

for each D ∈ D and ζ ∈ D. Then Ta : Ap → Ap is well-defined and bounded.Moreover, there exists a C ′ > 0 such that

‖Ta : Ap → Ap‖ ≤ C ′ supD∈D,ζ∈D

|aD(ζ)|.

If, in addition, we have

limd(D)→0

supζ∈D|aD(ζ)| = 0, (1.7)

then Ta is compact on Ap.

Note that the above theorem does not assume that a is positive and italso allows locally integrable symbols.

Berezin transform. One very important tool in the study of operators onBergman and Fock spaces is the Berezin transform. The idea is from thepaper [4] of F. A. Berezin. The Berezin transform is particularly powerfultool in the study of Toeplitz operators and elements in Toeplitz algebras. Leta ∈ L1(D), then the Berezin transform a of a can be defined as

a(z) =

∫D

(1− |z|2)2a(w)dA(w)

|1− zw|4. (1.8)

The integral kernel of the above operator is in fact |kz(w)|2 where kz(w) =Kz(w)/‖Kz‖2, as one can easily verify. For general p ∈ (1,∞), and an opera-tor T whose domain contains all of H∞ the definition for Berezin transformbecomes

T (z) = 〈Tk(q)z , k(p)

z 〉,where 1/p+ 1/q = 1,

k(p)z (w) =

(1− |z|2)2/q

|1− zw|2

6

and 〈·, ·〉 stands for the integral pairing of spaces Ap and Aq. We set a(z) =

Ta(z) and the new definition then agrees with (1.8) above when a ∈ L1. Thereader might want to consult [24] for more details on the general case, inparticular when not dealing with Hilbert space operators.

For a positive L1 symbol a, it is known that the Berezin transform of ais bounded if and only if a(z)dA(z) is a Carleson measure if and only if Ta isbounded on Ap for 1 < p <∞. Similarly, compactness can be formulated interms of vanishing of the Berezin transform near the boundary. These resultare proven in [37] for p = 2, for other values of p, these result follow fromthose in [26]. Also, N. Zorboska has improved this result for the space A2,see [38].

The Berezin transform is exceptionally useful when studying compact-ness of Toeplitz operators. In 1998 S. Axler and D. Zheng proved that fora ∈ L∞ the Berezin transform of a satisfies

lim|z|→1−

a(z) = 0

if and only if the Toeplitz operator Ta is compact on A2. This result wasproven for more general symbols by N. Zorboska in [38]. For our purposesthe most convenient result is due to D. Suarez, see [24].

Theorem 1.2.4. (Theorem 9.5 of [24]) Let 1 < p < ∞. Suppose that theoperator T is contained in the Banach algebra of operators on Ap generatedby all Ta with a ∈ L∞. Then T is compact on Ap if and only if

T (z)→ 0,

as |z| → 1−.

In the Fock space setting, the Berezin transform can be defined simi-larly, by inner product, or by using the integral formula like in (1.8). Therealso exists a more general concept, known as the heat-semigroup, which is apowerful tool for studying Toeplitz operators, see [3, 6].

1.2.1. The Bloch-type spaces and A1. By duality, the boundedness and com-pactness results for Toeplitz operators on A1 are closely related to those ofB. The Bloch space can be considered as an analytic version of BMO∂ , thespace of bounded mean oscillation in Bergman metric. The Bergman projec-tion is bounded BMO∂ → B, and therefore the question of boundedness canbe approached by finding the pointwise multipliers of BMO∂ . The readershould consult [34, 35] for reference. A general result for boundedness wasrecently obtained by Taskinen and Virtanen in [27].

Write W(z) = 1 − log(1 − |z|) for the logarithmic weight (a slightlydifferent logarithmic weight is used in papers [30] and [D], but actually bothweights will do here). Denote by X the set of locally integrable symbols asatisfying

|aD(ζ)| ≤ C/W(ζ)

for some C > 0 and all D ∈ D and ζ ∈ D (see (1.4) and (1.5)).

7

By BOlog we mean the logarithmically weighted space of bounded oscil-lation. A continuous function f : D→ C belongs to the space BOlog if thereexists a positive constant C such that

supw∈Bβ(z,1/2)

W(z)|f(z)− f(w)| ≤ C

for every z ∈ D.

Theorem 1.2.5. (Theorem 7 of [27]) Suppose a ∈ BOlog + X. Then Ta iswell-defined and bounded on A1.

Proposition 1.2.6. (Proposition 9 of [27]) Suppose a ∈ X and that

limd(D)→0

supD∈D,ζ∈D

|aD(ζ)|W(ζ) = 0,

then Ta is compact on A1.

Both of the results also hold for the classical Bloch space B, by duality.A different approach to the boundedness and compactness of Toeplitz

operators on the Bloch spaces Bd for d ∈ (0,∞) and A1 was given in 2006 byZ. Wu, R. Zhao and N. Zorboska. We only deal with B and A1, for detailsabout Bd with d 6= 1, we refer the reader to [30]. The results in their fullgenerality involve measures. We, however, only present the results under theassumption that a ∈ L∞ (for a general measure, the condition involves arather technical size estimate).

By the logarithmic Bloch space LB we mean the space of functions fanalytic on D satisfying

supz∈D

(1− |z|2) log (2/(1− |z|2))|f ′(z)| <∞.

Corollary 1.2.7. (Corollary 2.1 and Corollary 2.2 of [30]) Suppose a ∈ L∞.Then Ta : B → B is bounded if and only if P (a) ∈ LB. It follows thatTa : A1 → A1 is bounded if P (a) ∈ LB.

Analogous compactness results are also proven, see Theorem 3.1 of [30].

1.3. Fredholm theory

Fredholm operators appear often in applications and their properties are ofcrucial importance when studying integral equations or even abstract indextheory. Toeplitz operators possess a very rich Fredholm theory with beautifulresults on the index and essential spectra.

Recall that an operator T on a Banach space X is Fredholm if

n(T ) := dim kerT <∞ and d(T ) := dimX/TX <∞.The index of T is defined by the formula

IndT = d(T )− n(T ).

Equivalently, T is Fredholm if and only if it is invertible in the Calkin algebraof all bounded operators modulo compact operators; that is, there exists abounded operator S such that

TS = I +K1 and ST = I +K2

8

for some compact operators K1 and K2.

Hardy spaces. The Fredholm theory for Toeplitz operators in Hardy spacesis very well understood. An extensive treatment of this field is found in [8].Fredholm theory in Hardy spaces is also significantly different from Bergmanspaces. Reasons for this are the lack of effective factorization in Bergmanspaces and the general difficulty when dealing with the unit disk, instead ofthe unit circle, for instance. All of the results of this section are well-knownin the setting of Hardy spaces, when understood correctly.

Reflexive Bergman spaces. The most natural Bergman space on which Fred-holm theory can be established is the Hilbert space A2. Results in this di-rection are, among others, due to U. Venugopalkrishna and L. Coburn in[29] and [11], respectively. When letting 1 < p < ∞ we deal with reflexiveBergman spaces, where we still have a bounded Bergman projection, whichmakes the course of arguments much easier compared to the A1 theory.

Let us first consider the space A2. The algebra T (C) is the Banachalgebra generated by Toeplitz operators with symbols in C(D) (it is also aC∗-algebra of operators on A2).

A beautiful result of L. Coburn in [11] goes as follows:

Theorem 1.3.1. (Theorem 1 and Corollary to Theorem 1 of [11]) The algebraT (C) is irreducible and contains all compact operators. Each element T ∈T (C) satisfies

T = Ta +K (1.9)

for some a ∈ C(D) and compact operator K. Moreover, Ta is Fredholm if andonly if a(z) 6= 0 for all z ∈ T.

This result was generalized to reflexive Bergman space in 1992 by X.Zeng, see [31]. Zeng also showed that the commutator ideal of T (C) is equalto the ideal of compact operators.

The equation (1.9) is a consequence of the fact that for a, b ∈ C(D), thesemi-commutator Tab − TaTb is compact. The largest C∗-subalgebra of L∞

for which this is true was discovered by K. Zhu in his paper [32]. Let us callthis algebra Q. The main result by Zhu is the following.

Theorem 1.3.2. (Proposition 6 and Theorem 13 of [32]) Let a ∈ L∞. Thefollowing are equivalent:

(i) a ∈ Q;(ii) a ∈ VMO∂ ∩ L∞;

(iii) The Hankel operators Ha and Ha are both compact A2 → L2.

In particular, we have that Q = VMO∂ ∩ L∞.

In his paper Zhu also proves that if a ∈ Q, then the Berezin transformof a − a vanishes when approaching the boundary. In what follows, if a is afunction defined on D and 0 < r < 1, we will use the notation

ar := ar(θ) = a(reiθ).

9

If a is defined on D, we will also write a1(θ) := a(eiθ). By Ind ar (r ∈ (0, 1])wemean the winding number of the curve ar(θ) around zero. Since the Berezintransform of a ∈ Q behaves nicely, the following consequence now follows.

Theorem 1.3.3. (Theorem 15 of [32]) Let a ∈ Q. Then Ta : A2 → A2 isFredholm if and only if Ta is Fredholm if and only if there exists ε > 0 andr < 1 such that |a(z)| > ε, whenever r < |z| < 1. The Fredholm index of Tais then given by the formula

IndTa = − Ind aR,

where r < R < 1.

A description of the essential spectrum of Ta was also given; denoteby βD the Stone-Cech compactification of D (see [17]) and write a∗ for theunique extension of a to βD. We then have

σessTa = a∗(βD \ D).

In this thesis we will also consider the symbol class H := C(D)+H∞(D),the so-called Douglas algebra. The class H is a natural class in the Hardyspace setting, and gives a nice generalization for C(D) in the Bergman spacesetting, as well. An obvious difficulty, like in the case of Q, is then the lackof continuous boundary values.

The following theorem was proven in 1997 by B. R. Choe and Y. J. Lee.

Theorem 1.3.4. (Theorem 1 and Lemma 6 of [10]) Let 1 < p <∞ and a ∈ H.The operator Ta is Fredholm if and only if there exist ε > 0 and r < 1 suchthat |a(z)| > ε, whenever r < |z| < 1. The essential spectrum is given by

σessTa = a∗(βD \ D),

where a∗ is the unique extension of a to βD.

In their paper Choe and Lee also study Toeplitz operators on the unitball of Cn. The Fredholm criterion is analogous. They also prove that theFredholm index is always zero when n ≥ 2. An index formula for the casen = 1 is given in [E].

A natural class in the Bergman space setting is also the algebra ofpiece-wise continuous function. This involves joining the discontinuities withsuitable curves. We will not deal with piece-wise continuous symbols in thisthesis. A rather complete treatment of essential spectra and index theory inthe setting of A2 is given in [28]. To our knowledge, most of the questionsabout piece-wise continuous symbols are still open for Ap when p 6= 2.

Bergman space A1. When dealing with Toeplitz operators on the Bergmanspace A1, one immediately runs into one serious problem. The Bergman pro-jection is not bounded L1 → A1. It follows that a ∈ L∞ is not sufficient forthe boundedness of Ta. Conditions for boundedness are given in [25], [27] and[30], for instance.

10

The Fredholm theory for Toeplitz operators on A1 was studied in 2008by J. Taskinen and J. Virtanen. They establish Fredholmness criterion andindex-theorem for Toeplitz operators having symbols in

CL := C(D) ∩ VMO∂ log,

see [25] for more details. The following theorem, proven in [25], sums up thereason, why CL works fine in this setting.

Theorem 1.3.5. (Theorem 6, Corollary 8 and Theorem 10 of [25]) Supposea ∈ CL. Then Ta is bounded A1 → A1 and Ha is compact A1 → L1. If, inaddition, a(z) = 0 for each z ∈ T, then Ta is compact on A1.

The above theorem makes it possible to treat Ta : A1 → A1 in a fash-ion similar to the reflexive Bergman space case. The following theorem wasproven.

Theorem 1.3.6. (Theorem 11 and Theorem 12 of [25]) Suppose a ∈ CL. ThenTa : A1 → A1 is Fredholm if and only if a(z) 6= 0 for all z ∈ T. Moreover,the index of Ta is then given by the formula

IndTa = − Ind a1.

11

2. Properties of Toeplitz operators on analytic function spaces:from function symbols to distributions

This thesis consists of five articles. The following sections serve as short in-troductions to these articles. The order of the section is not the same as thechronological order of the respective articles. We made the choice to put thetheory of distributional symbols in the beginning, even though the paper [E]was actually the first one to be finished. The article [B], however, contains ashort section on Fredholm theory on the Bergman space A1. When present-ing results in the summaries of the articles, we always refer to results of theparticular article in question.

The notation in the article [E] agrees with the usual notation, which isused in the introductory part. In the other four articles, to avoid confusion,we will use the following conventions. Suppose that X and Y are normedspaces. For an element x ∈ X we denote by ‖x;X‖ the norm of x in X.Similarly, ‖T : X → Y ‖ is used for the operator norm of a bounded operatorT : X → Y .

2.1. Article A: Toeplitz operators with distributional symbols on Bergmanspaces

In this article we deal with the problem of extending the definition of Toeplitzoperators to distributional symbols. Our approach involves symbols thatare not necessarily compactly supported distributions. Some results aboutToeplitz operators with compactly supported distributional symbols can befound in [1]. We show how the developed machinery also goes to produce un-bounded function symbols that generate bounded (or even compact) Toeplitzoperators.

The symbol classesW−m,∞ν . Natural distributional symbol classes for Toeplitzoperators on Bergman spaces are the weighted distributional Sobolev spacesW−m,∞ν . Let m ∈ N. Then the weighted Sobolev space Wm,1

ν consists offunctions f measurable of D, for which the norm

‖f ;Wm,1ν ‖ :=

∑|α|≤m

∫D|Dαf(w)|ν(w)|α|dA(w) (2.1)

is finite. Here ν(z) = 1 − |z|2 is the standard weight and we use the multi-index-notation for α, |α| and Dα. It is easy to see that A1 ⊂ Wm,1

ν for eachm ∈ N.

Definition 2.1.1. Let m ∈ N. The weighted distributional Sobolev spaceW−m,∞ν consists of those distributions a ∈ D′ that admit a representation

a =∑|α|≤m

(−1)|α|Dαbα, (2.2)

where each bα belongs to L∞α ; that is, ‖bα;L∞α ‖ := ‖ν−|α|bα‖∞ <∞.

12

The derivative Dα in the above theorem is understood in the distribu-tional sense. We will equip W−m,∞ν with the norm

‖a;W−m,∞ν ‖ := inf max|α|≤m

‖bα;L∞α ‖,

where the infimum is taken over all possible representations (2.2). We provethe following:

Lemma 2.1.2. (Lemma 2.2 and Lemma 2.4) Let m ∈ N. Then the space ofcompactly supported test-functions C∞0 is dense in the Sobolev space Wm,1

ν .Moreover, the dual space of Wm,1

ν is isometrically isomorphic to W−m,∞ν

under the dual pairing

〈f, a〉 :=∑|α|≤m

∫DDαf(w)bα(w)dA(w) , f ∈Wm,1

ν , a ∈W−m,∞ν .

We note that the representation (2.2) is usually not unique, but eachrepresentation will give the same value for the dual pair 〈·, ·〉 above.

The distributional classes W−m,∞ν contain all the compactly supporteddistributions in the following sense. If a ∈ D′ is a compactly supported dis-tribution, then there exists m ∈ N such that a ∈ W−m,∞ν . This follows fromthe standard distribution theory, see for instance [22].

Boundedness and compactness of Toeplitz operators with distributional sym-bols. The definition for Toeplitz operator with symbol a ∈ W−m,∞ν goes asfollows.

Definition 2.1.3. Suppose a ∈W−m,∞ν for some m ∈ N. The Toeplitz operatorTa is then defined as

Taf(z) :=∑|α|≤m

∫DDα

(f(w)

(1− zw)2

)bα(w)dA(w) , z ∈ D. (2.3)

Here (bα) is a collection of functions satisfying (2.2).

The value of the above operator does not depend on the representationfor a. It also agrees with the standard definition for Toeplitz operator withL∞ symbol. The following two are the main theorems of this paper.

Theorem 2.1.4. (Theorem 3.1) Suppose a ∈ W−m,∞ν for some m ∈ N. Thenthe Toeplitz operator Ta is bounded on Ap for 1 < p < ∞. Moreover, thereexists C := C(m, p) > 0 such that

‖Ta : Ap → Ap‖ ≤ C‖a;W−m,∞ν ‖.

Theorem 2.1.5. (Theorem 4.2) Suppose a ∈ W−m,∞ν for some m ∈ N and ahas a representation

a =∑|α|≤m

(−1)|α|Dαbα,

where each bα satisfies

ess limr→1−

supr<|z|<1

ν−|α|(z)|bα(z)| = 0.

13

Then Ta is compact on Ap for 1 < p <∞.

2.2. Article B: New results and open problems on Toeplitz operators inBergman spaces

This article complements papers [A], [E], [25] and [26]. We also present openquestions related to Toeplitz operators on Bergman spaces.

Locally integrable symbols. We prove that if a ∈ L1loc is radial, then the

condition in [26] can be simplified. Suppose that a ∈ L1loc is radial: a(z) =

a(|z|). For all r ∈ (0, 1) let I := I(r) = [r, 1− (1− r)/2] and

aI(ρ) =1

1− r

∫ ρ

r

a(s)sds, (2.4)

where ρ ∈ I.The following lemma connects (2.4) to the main result of [26].

Lemma 2.2.1. (Lemma 2 and Corollary 3) Suppose a ∈ L1loc is radial. Then

a satisfies the condition (1.6) if and only if there exists C > 0 such that

supr∈(0,1)

supρ∈I(r)

|a(ρ)| ≤ C.

Moreover, if 1 < p <∞, then there exists Cp > 0 such that

‖Ta : Ap → Ap‖ ≤ Cp supr∈(0,1)

supρ∈I(r)

|a(r)|.

Similarly, we can establish a compactness result.

Lemma 2.2.2. (Lemma 8) For a radial a ∈ L1loc the condition

limr→1−

supρ∈I(r)

|a(ρ)| = 0 (2.5)

is equivalent with the vanishing condition (1.7).

It also follows that if a ∈ L1loc satisfies (2.5), then the associated Toeplitz

operator Ta is compact on Ap for 1 < p <∞.

Distributional symbols. We want to find a radial counterpart of the W−m,∞ν

classes defined in [A]. Let µ(r) = r(1 − r2) (r ∈ (0, 1)) be a radial weightfunction. Let m ∈ N. We can then define W−m,∞µ (0, 1) as the set of thosedistributions a on (0, 1) that have a representation

a =∑

0≤j≤m

(−1)jDjbj , (2.6)

where each bj satisfies ‖bj ;L∞j ‖ := ‖µ−jbj‖∞ <∞. We can makeW−m,∞µ (0, 1)a Banach space by equipping it with the norm

‖a;W−m,∞µ (0, 1)‖ = inf max0≤j≤m

‖bj ;L∞j ‖,

where the infimum is taken over all possible representations (2.6).It can be seen that for each m ∈ N we have W−m,∞µ (0, 1) ⊂ W−m,∞ν if

we understand bj(z) = bj(|z|). Hence every a ∈ W−m,∞µ (0, 1) can be seen togenerate a bounded Toeplitz operator on Ap for 1 < p <∞. We also prove:

14

Proposition 2.2.3. (Proposition 7) Suppose a ∈ L1loc is radial, a vanishes on

a neighbourhood of 0, and that there exists C > 0 such that

supr∈(0,1)

supρ∈I(r)

|a(ρ)| ≤ C.

Then there exists m ∈ N such that a ∈W−1,∞µ (0, 1).

There is a small mistake in the proposition; we need the additionalassumption that a vanishes on a neighbourhood of 0. This is actually notedin the proof and not at all that serious since we are mostly concerned aboutthe behaviour of a near the point 1.

A related compactness result can also be verified:

Proposition 2.2.4. (Theorem 9 and Proposition 10) Suppose that a ∈ L1loc is

radial, a vanishes on a neighbourhood of 0 and that a satisfies (2.5). Thena ∈W−1,∞

ν (0, 1) and has a representation

a =∑

j∈{0,1}

(−1)jDjbj ,

where

ess lims→1−

sups<r<1

µ(r)−j |bj(r)| = 0.

It follows that Ta is compact on Ap for 1 < p <∞.

Fredholm theory for matrix-valued symbols on A1. We discuss the Fredholmcriteria for matrix-valued symbols on (A1)N . This is an extension of theresults of [25] to the matrix-valued case. Denote byW(z) := 1+log(1/(1−|z|))the logarithmic weight on the unit disk. For a ∈ L1(D) and r ∈ (0, 1) we definethe r-mean-oscillation MOr(a) by

MOr(a)(z) =1

|B(z, r)|

∫B(z,r)

|a(w)− aB(z,r)|dA(w).

Here by aB(z,r) we mean the average of a over the Bergman disk B(z, r). Thevalue of r is not important, we can agree that r = 1/2 and just write MO(f).We say that a ∈ VMOlog if

lim|z|→1−

W(z)MO(f)(z) = 0.

We prove the following result, for more result about matrix-valued sym-bols, consult [E] and the corresponding section of this thesis.

Theorem 2.2.5. (Theorem 11) Let a be a matrix-valued symbol a = (ai,j)1≤i,j≤Nwith ai,j ∈ C(D) ∩ VMOlog. Then the Toeplitz operator Ta is Fredholm on(A1)N if and only if det a(z) 6= 0 on the boundary ∂D. In this case

IndTa = − Ind det(a)1.

15

2.3. Article C: Toeplitz operators with distributional symbols on Fock spaces

In this paper we establish in the Fock space setting results analogous to thosein [A]. Apart from dealing with symbols that are not necessarily functionsthis approach gives new results about unbounded function symbols. The ideabehind this paper is quite similar to that of [A]. However, some additional dif-ficulties emerge from the fact that the functions are defined on an unboundedset, the whole of C.

The symbol classes W−m,∞ω . Our aim is to find a suitable class of distribu-tions, much like in [A]. Let ω : C → R be the weight-function defined asω(z) = 1 + |z|. A reasoning similar to that of [A] leads us to consider thefollowing class of distributions.

Definition 2.3.1. Let m ∈ N. The weighted distributional Sobolev spaceW−m,∞ω consists of those distributions a ∈ D′ that admit a representation

a =∑|α|≤m

(−1)|α|Dαbα, (2.7)

where each bα belongs to L∞α ; that is, ‖bα;L∞α ‖ := ‖ω|α|bα‖∞ <∞.

The derivative Dα is understood in the distributional sense. We canmake W−m,∞ω a Banach space by using the norm

‖a;W−m,∞ω ‖ = inf max|α|≤m

‖bα;L∞α ‖, (2.8)

where the infimum is taken over all possible representations for a.

Integral estimates. The proof for the main result of this paper is analogousto that of [A]. However, there are some difficulties, which can be overcomewith the help of the following lemma.

Lemma 2.3.2. (Lemma 3.2, Corollary 3.3 and Lemma 3.4) Let k ∈ N anddefine the operators Tk, Sk and the mapping T ′k by

Tkf(z) = zk∫Cf(w)eγzwe−γ|w|

2

bα(w)dA(w) , z ∈ C;

T ′kf(z) =

∫Cω(z)kω(w)−k|f(w)eγzw|e−γ|w|

2

dA(w) , z ∈ C;

Skf(z) = ω(z)−kf (k)(z).

If |α| ≥ k and bα ∈ L∞α , we have ‖Tk : Lpγ → F pγ ‖ ≤ C1‖bα;L∞α ‖, ‖T ′kf ;Lpγ‖ ≤C2‖f ;Lpγ‖ and Sk is bounded F pγ → Lpγ .

Boundedness and compactness of Toeplitz operators with distributional sym-bols. The definition of Toeplitz operator in analogous to that of [A]. However,the reader should take notice how the Gaussian measure is taken into accountin the following definition.

16

Definition 2.3.3. Suppose a ∈W−m,∞ω for some m ∈ N. The Toeplitz operatorTa is then defined as

Taf(z) :=∑|α|≤m

∫CDα(f(w)eγzwe−γ|w|

2

)bα(w)dA(w) , z ∈ C. (2.9)

Here (bα) is a representation for a.

The boundedness and compactness theorems come out as follows.

Theorem 2.3.4. (Theorem 4.1) Suppose a ∈ W−m,∞ω for some m ∈ N. Thenthe Toeplitz operator Ta is bounded on F pγ for γ > 0 and 1 ≤ p ≤ ∞.Moreover, there exists C := C(m, p, γ) > 0 such that

‖Ta : F pγ → F pγ ‖ ≤ C‖a;W−m,∞ω ‖.

Theorem 2.3.5. (Theorem 5.2) Suppose a ∈ W−m,∞ω for some m ∈ N and ahas a representation

a =∑|α|≤m

(−1)|α|Dαbα,

where each bα satisfies

ess limr→∞

supr<|z|

ω|α|(z)|bα(z)| = 0.

Then Ta is compact on F pγ for γ > 0 and 1 ≤ p ≤ ∞.

2.4. Article D: Toeplitz operators on Bloch-type spaces and classes of weightedSobolev distributions

We study boundedness and compactness of Toeplitz operators acting betweenthe generalized Bloch-spaces of the unit disk. The article deals with operatorsBd → Bd′ with 0 < d, d′ <∞; to avoid repetition, this summary is only aboutToeplitz operators on the classical Bloch space.

Symbol classes LYm0 and LVm0 . When dealing with Bloch space operators,it is natural to consider logarithmically weighted symbol classes. We will usethe weight functions

ν(z) = 1− |z|2 and `(z) = 1 + | log(ν(z))|.

Definition 2.4.1. Let m ∈ N. The weighted distributional Sobolev space LYm0consists of those distributions a ∈ D′ that admit a representation

a =∑|α|≤m

(−1)|α|Dαbα, (2.10)

where each bα belongs to LL∞−|α|; that is, ‖bα;LL∞−|α|‖ := ‖`ν−|α|bα‖∞ <∞.

We note that in the actual paper we need several other distributionalclasses, in which the logarithmic weight might, or might not be present. How-ever, in each case the duality treatment and density results are similar tothose of [A, B, C].

To shorten the notation, we also introduce the classes LVm0 :

17

Definition 2.4.2. Let m ∈ N. A distribution a ∈ LYm0 is said to belong toLVm0 if a has a representation

a =∑|α|≤m

(−1)|α|Dαbα,

where each where each bα belongs to LL∞−|α| and

ess limr→1−

supr<|z|<1

`(z)ν(z)−|α||bα(z)| = 0.

The class LYm0 is just a logarithmic version of the class W−m,∞ν of [A].The class LVm0 implies behaviour similar to what happens in Theorem 2.1.5,for instance.

Boundedness and compactness of Toeplitz operators on the Bloch space.The definition for Toeplitz operator with symbol in LYm0 is the same as inDefinition 2.1.3. We list here the main results for Ta : B → B.

Theorem 2.4.3. (Part (2) of Corollary 3.8) Suppose a ∈ D′ belongs to LYm0for some m ∈ N. Then Ta is bounded on the Bloch space and there exists aC := Cm such that

‖Ta : B → B‖ ≤ C‖a;LYm0 ‖.

Theorem 2.4.4. (Part (2) of Corollary 4.7) Suppose a ∈ D′ belongs to LVm0for some m ∈ N. Then Ta is compact on the Bloch space.

It can be shown that the symbol class LYm0 fails to contain constantsfunctions, which is naturally a serious drawback, since a Toeplitz operatorwith contant symbol is just a multiple of the (obviously bounded) identityoperator. This inconvenience can be partially fixed by studying logarithmicBMO∂ .

Let f ∈ L∞. Denote by f the averaging function:

f(z) = |B(z, r)|−1

∫B(z,r)

f(w)dA(w),

where B(z, r) is a Bergman disk.In what follows, the choice of r ∈ (0, 1) is not important; just agree that

r = 1/2, for instance.

Definition 2.4.5. Let f ∈ L∞. We say that f belongs to the logarithmicBMO∂ (f ∈ BMOlog) if

‖f ;BMOlog‖ := supz∈D

`(z)|D(z, r)|−1

∫D(z,r)

|f(w)− f(z)|dA(w) <∞.

We manage to improve our main Theorem 2.4.3 by the following.

Theorem 2.4.6. Let a ∈ D′ be a member of Ym0 for some m ∈ N. Assumemoreover, that there exists m′ ∈ N such that a has a representation

a =∑|α|≤m′

Dαbα,

18

where bα/ν|α| ∈ BMOlog for each α. Then Ta is bounded on B.

The Bergman space A1. By duality, we also establish theory of distributionalsymbols for the Bergman space A1.

Corollary 2.4.7. Suppose a ∈ D′ belongs to LYm0 for some m ∈ N. Then Tais bounded on A1 and there exists a constant C := Cm > 0 such that

‖Ta : A1 → A1‖ ≤ C‖a;LYm0 ‖.

Corollary 2.4.8. Suppose a ∈ D′ belongs to LVm0 for some m ∈ N. Then Tais compact on A1.

Corollary 2.4.9. Let a ∈ D′ be a member of Ym0 for some m ∈ N. Assumemoreover, that there exists m′ ∈ N such that a has a representation

a =∑|α|≤m′

Dαbα,

where bα/ν|α| ∈ BMOlog for each α. Then Ta is bounded on A1.

2.5. Article E: A note on the Fredholm properties of Toeplitz operators onweighted Bergman spaces with matrix-valued symbols

We deal with Fredholm theory for Toeplitz operators having matrix-valuedsymbols. There are several results on the Hardy space setting, but the Bergmanspace theory seems still relatively incomplete. The reason might be, that ex-cept for some simple classes (such as C(D)), the reduction of the matrix-valued case to the scalar-valued case is more complicated. Probably thebest reference for the matrix-valued symbols on Hardy spaces is [8]. For theBergman space we refer the reader to [11].

The weighted Bergman spaces of the unit ball. We work on the weightedBergman spaces Apα(Bn) on the unit ball of Cn with α > −1 and p ∈ (1,∞).The Bergman projection Lpα(Bn)→ Apα(Bn) is then defined as

Pαf(z) =

∫Bn

f(w)dAα(w)

(1− 〈z, w〉)n+1+α,

where dAα(z) = cα(1 − |z|2)αdA(z), dA is the standard 2n-dimensionalLebesgue measure and cα is a constant which makes dAα a probability mea-sure.

If a ∈ L∞(Bn), then the Toeplitz operator Ta is defined as

Taf(z) = Pα(af) , f ∈ Apα.

Scalar-valued symbols. The symbols classes under consideration are C :=C(Bn), H := C(Bn) + H∞(Bn) and Q := VMO∂(Bn) ∩ L∞(Bn) and theirmatrix-valued counterparts: CN := CN×N , HN := HN×N and QN := QN×N ,respectively. We will omit the dimension n from the notation; it is always thesame as the n in Apα(Bn). Both n and N are assumed to be natural numbersthroughout this section. The Fredholm conditions for scalar-valued C and Hare known, see for instance [10, 31] and the section of this thesis dealing withFredholm theory. In our paper, we prove:

19

Theorem 2.5.1. (Theorem 3) Let 1 < p < ∞, n = 1, α > −1 and a ∈ H.Then Ta is Fredholm on Apα(D) if and only if a is bounded away from zeronear the boundary; that is, there exists ε > 0 and r ∈ (0, 1) such that

|a(z)| > ε , when r < |z| < 1.

Moreover, the index of Ta then satisfies

IndTa = − Ind aR,

for all R ∈ (r, 1).

We recall that for n > 1, the index of Ta is known to be zero whenevera ∈ H and Ta is Fredholm. We establish a similar result for a scalar-valueda ∈ Q. This is achieved by using the Berezin transform and the results in[32, 24].

Theorem 2.5.2. (Theorem 5) Let 1 < p <∞, α > −1 and a ∈ Q. Then Ta isFredholm on Apα(Bn) if and only if the Berezin transform of a is bounded awayfrom zero near the boundary of Bn; that is, there exists ε > 0 and r ∈ (0, 1)such that

|a(z)| > ε , when r < |z| < 1.

If the above condition is satisfied and n > 1, then the index of Ta is zero; ifn = 1, then the index of Ta satisfies

IndTa = − Ind aR,

for all R ∈ (r, 1).

Matrix-valued symbols. Theorems concerning matrix-valued symbols can of-ten be reduced to theorems about scalar-valued symbols. In the setting ofHardy spaces, this is again settled, see [8]. The Bergman space theory ismore difficult, mostly due to lack of boundary-values and the behaviour ofthe commutator TaTb−TbTa. The criteria for Fredholmness are still possibleto verify:

Theorem 2.5.3. (Theorem 7) Let 1 < p <∞, N ≥ 2 and α > −1.

(i) If a ∈ CN , then Ta is Fredholm on (Apα(Bn))N if and only if det a(z) 6= 0on ∂Bn; if in addition n = 1, we have

IndTa = IndTdet a = − Ind(det a)1.

(ii) If a ∈ HN , then Ta is Fredholm on (Apα(Bn))N if and only if det a(z) isbounded away from zero near the boundary.

(iii) If a ∈ QN , then Ta is Fredholm on (Apα(Bn))N if and only if A(z) isbounded away from zero near the boundary, where A(z) = det a(z).

We were unable to verify the index-formula for the classes HN and QNif N 6= 1. We managed to prove a partial result:

Theorem 2.5.4. (Theorem 8) Let 1 < p <∞, N ≥ 2 and α > −1 and n = 1.Suppose that either a ∈ HN or a ∈ QN , Ta is Fredholm on (Apα(D))N andthat at least one of the following holds:

20

(i) The scalar-valued Toeplitz operators Tai,j and Tak,l 1 ≤ i, j, k, l ≤ Ncommute modulo trace class operators. Here a = (ai,j).

(ii) The operator Tak is Fredholm on (Apα(D))k for each 1 ≤ k ≤ N , whereak = (ai,j)1≤i,j≤k.

Then the index formulaIndTa = IndTdet a

holds.

21

3. Notes

3.1. Complementary material

The definition for weighted Sobolev spaces of distributions works well if onealready has a suitable representation for a distribution a ∈ D′. However,in general such representation might be problematic to find. The followingproposition gives another method by the use of duality. We formulate theresult only for the symbol class Ym0 ; a similar result is true for all the distri-butional Sobolev classes presented in this thesis.

Proposition 3.1.1. (Proposition 2.5 of [D]) Suppose a ∈ D′. Then a ∈ Ym0 ifand only if there exists a positive constant C such that

|〈ϕ, a〉| ≤ C‖ϕ;Wm,1ν ‖

for every compactly supported test function ϕ.

For a compactly supported distribution a it is natural to define theToeplitz-type operator Sa by the formula

Saf(z) = 〈f(w)(1− zw)−2, a〉w.

One might argue whether this definition is consistent with Definition 2.1.3given in this thesis. We have shown the following (again, the same is true forevery Sobolev class under consideration):

Proposition 3.1.2. (Proposition 4.2 of [D]) Suppose a is a compactly supporteddistribution. Then there exists an m ∈ N such that a ∈ Ym0 . Moreover

Sa = Ta,

in particular the operator Sa is a Toeplitz operator in the sense of Definition2.1.3.

We would also like to note that, in the Fock space setting, it is possibleto define a Toeplitz operator Ta, where a ∈ S ′ is a tempered distribution.The definition is given by the formula

Taf(z) = 〈f(w)eγzwe−γ|w|2

, a〉w,

where 〈·, ·〉 stands for the dual bracket in 〈S,S ′〉. The definition is then moregeneral than ours, but boundedness and compactness questions become morecomplicated.

3.2. Errata

There is a mistake in proposition 7 of [B]; an additional assumption on van-ishing of a on a neighbourhood of 0 is needed. This mistake is not that serioussince this only means perturbating the operator by a compact operator, thushaving no effect on boundedness.

The example 1 of [B] is also slightly misleading; it would be more correctto assume that h is a linear form on the space of real-analytic functions on D,instead of being a (possibly discontinuous) linear form on C∞. The example

22

presented is, in fact, of the former type and correct.

A misprint appears a couple of times in article [D]. The proofs of Propo-sition 4.2, Lemma 4.3 begin with ”Proofs of Theorems 3.5-3.7”, which is in-correct, as the proofs are those of the aforementioned proposition and lemma.Similarly, the proofs of Lemma 5.3 and Theorem 5.4 begin with ”Proof ofTheorems 4.4-4.6”.

Also, in page 114 of [D], the chain of inclusions is not correct. The right-most space should be A1

d−1; this claim does not hold, in general, for p > 1.This mistake has no effect on the outcome of the paper.

In [E] we prematurely conjecture that there should exist symbols a andb both polynomials of z and z such that the commutator

TaTb − TbTais not trace class on A2. This conjecture was later proven false by TrieuLe, who contacted the author on this matter. The author wishes to thankProfessor Le for pointing this out.

23

References

[A] A. Perala, J. Taskinen, J. Virtanen, Toeplitz operators with distributional sym-bols on Bergman spaces. Proceedings of the Edinburgh Mathematical Society,Volume 54, Issue 02, 505–514, (2011).

[B] A. Perala, J. Taskinen, J. Virtanen, New results and open problems on Toeplitzoperators in Bergman spaces. New York Journal of Mathematics, Volume 17a,147–164, (2011).

[C] A. Perala, J. Taskinen, J. Virtanen, Toeplitz operators with distributional sym-bols on Fock spaces. Functiones et Approximatio, Commentarii Mathematici,Volume 44, Number 2, 203–213, (2011).

[D] A. Perala, Toeplitz operators on Bloch-type spaces and classes of weightedSobolev distributions. Integral Equations and Operator Theory, Volume 71,Number 1, 113–128 (2011).

[E] A. Perala, J. Virtanen, A note on the Fredholm properties of Toeplitz opera-tors on weighted Bergman spaces with matrix-valued symbols. Operators andMatrices, Volume 5, Issue 1, 97–106, (2011).

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